Next consider the entropy function \(H(t) = t \cdot {\rm log}(t) - t \) ,
whose Legendre-Fenchel transform equals \(H^*(s) = {\rm exp}(s)\) .
Here, (REF ) is the entropic regularization of (REF ).
This approach is popular in machine learning, especially for
optimal transport problems [1]}, [2]}, [3]}.
Note that \(H(t)\)  is strictly convex but not strongly convex. It is not a barrier function since
\(H(t)\)  does not diverge for \(t \rightarrow 0\) .
But, its derivative does, and this ensures the minimizer \(x^*(\epsilon )\)  to be in the
relative interior of \(P_{A,b}\) .
To highlight algebraic features, we assume
that the cost vector \(c\)  has integer coordinates.
The rank condition in (REF ) is a system of \(\mathbb {Z}\) -linear equations
in \({\rm log}(x_1), \ldots , {\rm log}(x_n)\) .
These translate into
differences of monomials in \(\mathbb {R}[x_1,\ldots ,x_n]\) .
Indeed, (REF )
specifies the toric variety of the integer matrix \(\binom{A}{c}\) .
We obtain the entropic
curve \(\mathcal {C}^T_{A,b,c}\)  by
intersecting that toric variety
with the linear space \(\lbrace Ax = b\rbrace \) .
Its degree is bounded by the normalized volume of a polytope
given by \( \binom{A}{c} \) , and
\(x^*(\infty )\)  is the Birch point of \(P_{A,b}\) .
